Georgia Institute of Technology College of Sciences

TBA

We develop a unified framework for improving numerical solvers with Neural Networks with Locally Converging Inputs (NNLCI). First, we applied NNLCI to 2D Maxwell’s equations with perfectly matched‐layer boundary conditions for light–PEC (perfect electric conductor) interactions. A network trained on local patches around specific PEC shapes successfully predicted solutions on globally different geometries. Next, we tested NNLCI on various ODEs: it failed for chaotic systems (e.g., double pendulum) but was effective for nonchaotic dynamics, and in simple cases can be interpreted as a well‐defined function of its inputs. Although originally formulated for hyperbolic conservation laws, NNLCI also performed well on parabolic and elliptic problems, as demonstrated in a 1D Poisson–Nernst–Planck ion‐channel model. Building on these results, we applied NNLCI to multi‐asset cash‐or‐nothing options under Black–Scholes. By correcting coarse‐ and fine‐mesh ADI solutions, NNLCI reduced RMSE by factors of 4–12 on test parameters, even when trained on a small fraction of the parameter grid. Careful treatment of far‐field boundary truncation was critical to maintain convergence far from the strike price. Finally, we demonstrate NNLCI’s first application to quantum algorithms by improving variational quantum‐algorithm (VQA) outputs for the 1D Poisson equation under realistic NISQ‐device noise. Although noisy VQA solutions deviate from classical finite‐difference references and do not converge to true solutions, NNLCI effectively maps these noisy outputs toward high‐accuracy references. We hypothesize that NNLCI implicitly composes the map from coarse quantum outputs to a noisy convergence space, then to the true solution. We discuss conditions for NNLCI to approximate a well‐defined inverse of the numerical scheme and contrast this with Monte Carlo methods, which lack deterministic intermediate states. These results establish NNLCI as a versatile, data‐efficient tool for accelerating solvers in classical and quantum settings.

A knot in a contact manifold is Legendrian if it is everywhere tangent to the contact planes. The classification problem in Legendrian knot theory has always generated significant interest. The problem gets a lot more complicated when we consider links. In this talk, I'll survey some of the results in this area and then discuss the classification problem for cable links of uniformly thick knot type.  If time permits, I'll also mention the classification of links in the overtwisted setting. Part of this is joint work with John Etnyre, Hyunki Min, and Tom Rodewald. 

This thesis explores Turán-type extremal problems in graphs that exclude certain minors, focusing on the maximum number of $k$-cliques such graphs can contain. The first part of the thesis studies planar graphs, which forbid $K_5$ and $K_{3,3}$ as minors. We determine the maximum number of edges is in a planar graph that contains no cycle of length k, and establish a general upper bound for the number of edges in a planar graph avoiding $C_k$ for any $k\ge 11$.

The second part addresses the maximum number of $k$-cliques in $K_t$-minor-free graphs. We show essentially sharp bounds on the maximum possible number of cliques of order $k$ in a $K_t$-minor-free graph on $n$ vertices. More precisely, we determine a function $C(k, t)$ such that for each $k < t$ with $t - k \gg \log_2 t$, every $K_t$-minor-free graph on $n$ vertices has at most $n \cdot C(k, t)^{1 + o_t(1)}$ cliques of order $k$. We also show that this bound is sharp by constructing a $K_t$-minor-free graph on $n$ vertices with $C(k, t) n$ cliques of order $k$. This result answers a question of Wood and Fox–Wei asymptotically up to an $o_t(1)$ factor in the exponent, except in the extreme case where $k$ is very close to $t$.